When are there not min/max values of a function subject to a constraint?

Question

How do I know if there are no extreme values of a function subject to a constraint? For example, if $f(x,y,z)=xy+3xz+2yz$ subject to the constraint $5x+9y+z=10$. Why does it not have min/man values?

Answer 1

Given a continuous function $f:E\subset \Bbb{R}^n \rightarrow \Bbb{R}$, if $E$ is closed and bounded (that is, compact) then by the Weiestrass theorem we know that there exists maximum and minimum values for the function on $E$.

If $E$ is not closed or not bounded (or neither), or the function is not continuous, then we can't say anything (at least right away) regarding the existence of minimum and maximum values. It might have a maximum, or a minimum, or both, or none. Anything could happen.

In your particular case, the constraint $5x+9y+z=10$ is a plane which is clearly unbounded, so we can't say anything about the existence of global extrema.

Now note that $f$ can be made arbitrarily small:

We have that $\left (\frac{10-9y}{5},y,0 \right ) \in M$ for all $y \in \Bbb{R}$ where M is the plane with equation $5x+9y+z=10$. Then:

$$f\left (\frac{10-9y}{5},y,0 \right ) =\frac{10-9y}{5}y \rightarrow -\infty \ as \ y \rightarrow +\infty$$

Hence $f$ does not have a minimum.

On the other hand, since $z=10-9y-5x$ we get:

$f(x,y,z)=-36xy+30x-15x^2+20y-18y^2$

Now this tends to $-\infty$ as $x$ and $y$ tend to $+\infty$ or $-\infty$ so it would seem that there is in fact some global maximum. There is probably a more rigorous way of proving that such a maximum exists but I can't think of it right right now. If I come up with anything I'll edit my answer.

Answer 2

The problem that you're facing is that the variables can vary from $-\infty$ to $\infty$. Set one of the variables to 0, the other to some big negative number, let's say -1000000, and you'll get a number of simular size for the third one. Then you $f(x,y,z) \to -\infty$ as you get a bigger and bigger negative number (in absolute value sense). Simularyl you can get that $f(x,y,z) \to \infty$ for some other number, hence the function is unbounded, which is mainly to the fact that the variables are not bounded.

On the other side if the variables were bounded, for example $x,y,z>0$ I'm pretty sure you can find the extreme points for the fundtions